Topo-isomorphisms of irregular Toeplitz subshifts for residually finite groups

For each countable residually finite group $G$, we present examples of irregular Toeplitz subshifts in $\{0,1\}^G$ that are topo-isomorphic extensions of its maximal equicontinuous factor. To achieve this, we first establish sufficient conditions for Toeplitz subshifts to have invariant probability measures as limit points of periodic invariant measures of $\{0,1\}^G$. Next, we demonstrate that the set of Toeplitz subshifts satisfying these conditions is non-empty. When the acting group $G$ is amenable, this construction provides non-regular extensions of totally disconnected metric compactifications of $G$ that are (Weyl) mean-quicontinuous dynamical systems.